I'm unsure how to decide whether the normal should be positive or negative in $\hat{n}dS=\pm h_2 h_3 e_1 du_2 du_3$, where $h_i$ are the scale factors, $e_i $ are the base vectors, and $u_i$ are the ...

I have stumbled upon an exercice for second year undegraduate student majoring in economics which I find quite demanding. I have an idea for the solution, but it seems awfully complicated, and I am ...

I understand how certain matrices can be orthonormal and the conditions necessary for that. I don't understand its geometric relevance. For instance in a vector space the axis would be orthonormal and ...

I have a confusion as to why this is a viable procedure in the following image:
$$
\sum_{n} B_n \int_{0}^{a} \sin \left( \frac{n \pi x}{a}\right) \sin \left( \frac{m \pi x}{a}\right)
= \sum_{n} B_n \...

Suppose there exist a subset $M$ of an inner product space $X$, and the orthogonal complement of $M $ is the zero vector. If $X $ is a Hilbert Space then the span of $M $ will be dense in $X $, but ...

Question
I am completely lost on this problem. I know how to find it using Gram-Schmidt but I'm unsure of how to even find the subspace in this case, or how I would graph any of this. Is there another ...

I have learned that a symmetric matrix must be able to be written in form of $S=QDQ^T$ where Q is the orthonormal eigenvectors. But I saw an example that display a symmetric matrix in the form of $S = ...

I have $k$ vectors $v_i\in\mathbb{R}^n$, mutually orthogonal. I would now like to rotate them in the $k$-dimensional subspace spanned by the $v_i$ such that $v_0$ ends up at the given target vector $w\...

Let $A(t) = \{x_1(t), x_2(t),..., x_n(t) \}$ with $0 \leq t \leq 1$ where $x_i(t) \in \mathbb{R}^n, \forall i$.
I would like to construct a set $A(t)$ such that
$A(t) $ is an orthonormal set, i.e. $...

I have to construct a diagonalizable and orthogonal matrix starting from this quadratic form $Q(x_1,x_2,x_3)=-2x_1x_3+2x_1x_3-2x_2x_3$ in order to reduce it in canonic form with a variables change.
I'...

Let $H$ be a Hilbert space and let ${e_n} ,\ n=1,2,3,\ldots$ be an orthonormal basis of $H$. Suppose $T$ is a bounded linear oprator on $H$. Then which of the following can not be true?
$$(a)\quad T(...

Our problem is to compute the eigenvalues and eigenvectors of two matrices formed by products of orthonormal vectors, $\mathbf{u}$ and $\mathbf{v}$:
$A=\mathbf{u}\mathbf{v}^T+\mathbf{v}\mathbf{u}^T$
...